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A283368 Irregular triangle read by rows: T(n,k) = number of heights for the horizontal elements of the Dyck paths for the symmetric representation of sigma(n) that are listed in the corresponding positions of the triangle of A259176. 2

%I

%S 1,2,3,2,4,3,5,3,6,5,4,7,5,4,8,6,5,9,7,5,10,8,7,6,11,8,7,6,12,10,9,7,

%T 13,10,9,7,14,11,9,8,15,12,11,10,8,16,13,12,11,9,17,13,12,11,9,18,15,

%U 13,12,10,19,15,13,12,10,20,16,15,13,11,21,17,16,15,14,11

%N Irregular triangle read by rows: T(n,k) = number of heights for the horizontal elements of the Dyck paths for the symmetric representation of sigma(n) that are listed in the corresponding positions of the triangle of A259176.

%C The dot product of the n-th row of this triangle and the n-th row of triangle A259176 equals A024916(n), the sum of all divisors of numbers 1 through n (true for all n <= 20000); the value is the sum of the rectangles between the x-axis and the horizontal legs of the symmetric representation of sigma(n). This is the companion computation to A283367.

%F T(n,k) = n - sum_{i=1..k-1} f(n, 2*i) where f is defined in A237593.

%F A024916(n) = sum_{i=1..row(n)} (T(n,i))*S(n,i)) where S(n,i) refers to the triangle of A259176 and row(n) = floor(sqrt(8*n+1)-1)/2).

%e The first horizontal leg of the symmetric representation of sigma(15) is at y-coordinate 15 and has length 8, and row 15 has 5 entries so that T(15,1) = 15 and T(15,5) = 8.

%e The first 16 rows of the irregular triangle:

%e 1

%e 2

%e 3 2

%e 4 3

%e 5 3

%e 6 5 4

%e 7 5 4

%e 8 6 5

%e 9 7 5

%e 10 8 7 6

%e 11 8 7 6

%e 12 10 9 7

%e 13 10 9 7

%e 14 11 9 8

%e 15 12 11 10 8

%e 16 13 12 11 9

%t (* function f[n,k] and its support functions are defined in A237593 *)

%t a283368[n_, k_] := n - Sum[f[n, 2i], {i, k-1}]

%t TableForm[Table[a283368[n, k], {n, 1, 16}, {k, 1, row[n]}]] (* triangle *)

%t Flatten[Table[a283368[n, k], {n, 1, 21}, {k, 1, row[n]}]] (* sequence data *)

%Y Cf. A024916, A237593, A259176, A259177, A283367.

%K nonn,tabf

%O 1,2

%A _Hartmut F. W. Hoft_, Mar 06 2017

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Last modified September 26 20:34 EDT 2021. Contains 347672 sequences. (Running on oeis4.)